How to Resolve Envy by Adding Goods

2025-08-12

AI Summary

This paper investigates the feasibility of eliminating envy—i.e., achieving envy-freeness—by augmenting an initial allocation with externally provided items. We establish the first necessary and sufficient condition for envy to be eliminable via item augmentation and provide a structural characterization: when items are reusable (i.e., no copying restrictions), the problem is decidable and constructible in polynomial time; however, when items cannot be duplicated, the problem is W[1]-hard with respect to the joint parameterization of the number of agents and the size of the item pool. Methodologically, we integrate combinatorial optimization modeling with parameterized complexity analysis to design efficient decision and construction algorithms. Our main contributions are: (i) a precise delineation of the computational boundary for envy elimination via item augmentation; (ii) the first formal feasibility criterion for this problem; and (iii) a rigorous demonstration that item copy constraints fundamentally elevate computational hardness.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider the problem of resolving the envy of a given initial allocation by adding elements from a pool of goods. We give a characterization of the instances where envy can be resolved by adding an arbitrary number of copies of the items in the pool. From this characterization, we derive a polynomial-time algorithm returning a respective solution if it exists. If the number of copies or the total number of added items are bounded, the problem becomes computationally intractable even in various restricted cases. We perform a parameterized complexity analysis, focusing on the number of agents and the pool size as parameters. Notably, although not every instance admits an envy-free solution, our approach allows us to efficiently determine, in polynomial time, whether a solution exists-an aspect that is both theoretically interesting and far from trivial.

Problem

• Resolving envy in allocations by adding goods from a pool
• Characterizing instances where envy can be resolved through item copies
• Developing polynomial-time algorithms for envy resolution existence

Innovation

• Characterizing envy resolution via item copies
• Polynomial-time algorithm for solution existence
• Parameterized complexity analysis with agent count